Primality proof for n = 9311:

Take b = 2.

b^(n-1) mod n = 1.

19 is prime.
b^((n-1)/19)-1 mod n = 4238, which is a unit, inverse 8720.

7 is prime.
b^((n-1)/7)-1 mod n = 763, which is a unit, inverse 8530.

(7^2 * 19) divides n-1.

(7^2 * 19)^2 > n.

n is prime by Pocklington's theorem.